A numerical study on the relationship between the doping and performance in P3HT:PCBM organic bulk heterojunction solar cells

In this study, we perform a simulation analysis to investigate the influence of p-type and n-type doping concentration in BHJ SCs using the drift-diffusion model. Specifically, we investigate the effect of doping on the charge carrier transport and calculate the above-mentioned device parameters. We show that doping the active layer can increase the cell characteristic parameters, that the results are in an excellent agreement with the experimental results previously reported in the literature. We also show that doping causes space charge effects which subsequently lead to redistribution of the internal electric field in the device. Our results reveal that higher doping levels lead to screening the electrical field in the P3HT:PCBM active region. This in turn forces the charge carrier transport to be solely dominated by the diffusion, consequently decreasing the performance of the device. We also show that doping of the active layer to an optimum level can effectively improve the charge transport. Moreover, we show that doping can create an Ohmic contact between the organic and cathode interface. Additionally, the charge carrier concentration profile shows that by increasing the dopant concentration, the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_{sc}$$\end{document}Jsc can be improved remarkably. Upon doping the active layer, this indicates that illumination can simply reduce the series resistance in the device.

In an organic BHJ SC, the hole-transporting polymer is P3HT and the photogenerated electrons are transferred through the electron-transporting PCBM polymeric channels.
Recombination in the active layer is governed by different mechanisms such as bimolecular (or Langevin recombination), Shockley-Read-Hall (SRH) recombination, trap assisted, or geminate recombination. We have taken these models into account in our numerical simulations.

A. Bimolecular recombination
Bimolecular (or Langevin recombination), R0, with the recombination constant (β) and the intrinsic charge carrier density (ni), which is governed by the electron (n) and hole (p) densities is given by the following equation: where β is a function of the electron and hole mobilities (µn and µp): where εε0 is the permittivity of the active layer P3HT:PCBM blend material, and q is the elementary charge. µn and µp are electron and hole mobilities, respectively. For undoped functional materials in the active layer, both electrons and holes are transported through the same material in the present work, we have used unbalanced mobilities of electrons and holes in our calculations.

B. Charge Transferred (CT) recombination
Photogeneration of the free charge carriers has been explained by the Onsager theory [5], and Braun [6] has made an important refinement to this theory by pointing out that a bound e-h pair with binding energy EB -which acts as a precursor for free charge carriers-has a finite lifetime.
EB is considered as an intermediate state through which the recombination and dissociation of charge carriers are triggered. The charge carriers in an e-h pair can return to their initial states or dissociate into free charge carriers. This charge carrier dissociation is a competition between the separation rate, kdiss, and the lossy radiative or non-radiative recombination of charge carriers which undergo transferring through an intermediate state, that is, the charge-transfer (CT) state. In this model the probability of dissociation for a given e-h pair distance x, is given by: In this equation, kdiss depends on both temperature, T, and electric field strength, E. The decay rate of the bound e-h pair to the ground state, kf, is used as the fitting parameter. Based on the Onsager theory for a field-dependent dissociation rate constant in the case of weak electrolytes with low mobility [1,5], Braun derives the following expression for kdiss: ), and R is the recombination rate.
As polymer systems in BHJ SCs are subject to disorder, it is reasonable to assume that the e-h pair distance is not constant throughout the system [1]. As a result, it should be integrated over the distribution of separation distances: where f(a,x) is a normalized distribution function that is given by [3]: This leads to a modification of free charge carrier recombination terms by P that describes the probability of the dissociation of a CT state, and consequently, geminate recombination is determined as [1,4]: where Rg is the geminate recombination, respectively.

C. Shockley-Read-Hall recombination
Another mechanism that can account for losses in organic BHJ SCs is the indirect recombination which nd and p0 are characteristic charge carrier densities. For simplicity, we assume a temperature-independent effective bandgap.

D. The model
The basic equations used in this paper are the Poisson equation: where E(x) is the applied electric field, Nn and Np are n-type and p-type doping concentrations, respectively. The current continuity equations are: In these equations, Jn, Jp are the electron and hole current densities, G is the generation rate of free charge carrier result from the separation of e-h excitons, and R is the recombination rate [10]. To calculate the exciton density profile, we can use:  4 where S, τs, γns and γps are the exciton density, lifetime of excitons, and the second order rate constant for the annihilation of excitons into free electrons and holes, respectively [10].
Considering that the total produced charge carriers recombine inside the BHJ, we can ignore the current gradient. We solve these equations via the FEM method by applying the boundary conditions and assuming a typical 100 nm thick active layer. The scheme used to solve the Poisson and continuity equations is based on the work of Gummel [9]. This new potential is then used to update the carrier densities by solving the continuity equations [6,9].
where Pin is the incident optical power.